Well, theoretically you could have as many patriarchates as you like. The patriarchal system was originally set up within the Roman Empire (and it developed over time). At first it was an administrative measure: in each region of the empire the bishops would have local councils/synods, and these would meet in cities of administrative importance (or, in the case of Jerusalem, symbolic importance). The 'patriarch' was the bishop of the city in which the synod met, and was effectively considered to be a chairman, not a monarch. After a while the Church came to be identified as a sort of ecclesiastical parallel to the Roman Empire - one was universal in religious terms, the other in political terms. As the Roman Empire broke down and the 'Byzantine' Empire took shape, the Patriarchate of Constantinople came to be identified with the boundaries of Byzantium. Even so, the patriarchates of Rome, Alexandria, Antioch and Jerusalem were still seen as being united with Constantinople, and were all under the general political sway of Byzantium. In the 10th and 11th centuries the Papacy came under the sway of the German Holy Roman Empire, and with the papal reform movement it came to distinguish itself from all other patriarchates. Eventually the Papacy began to insist that the Pope was the sole ruler of the Church, just as the German Emperors were claiming to be the sole rulers of the Roman Empire. However, the other Patriarchates were all still under the influence of Byzantium and stayed united with one another.Originally Posted by RedRooster81
It's important to understand the subtle relationship of the Orthodox patriarchates to the Byzantine imperial throne. Unlike in the west, the Byzantine emperor was generally strong enough to stop any patriarch from developing pretensions to political power (as the Roman popes did). However, a patriarchate was a mark of political power for secular rulers, since only great administrative/imperial cities were supposed to have one. Thus, when the Bulgarians created a patriarchate, it was seen as a statement of political indepedence - they weren't reliant on the Byzantine emperor any more.
As for the creation of new patriarchates in places like Baghdad, it's a hard question to answer. In purely ecclesiastical terms, there is no set number of patriarchates - you have as many as is necessary. In political terms, it might not be so desirable. Ultimately the Byzantines never conquered Baghdad, so it's impossible to say what they would have done.